Contents

Examples

Let's start with an easy and well-known summation.

Find a general formula for \(1 + 2 + 3 + \dots + n.\)

Let's start by assuming that the above summation can be expressed as a polynomial function \(f\left( n \right)\) of the form
\[f\left( n \right) = { A }_{ 0 }+{ A }_{ 1 }n+{ A }_{ 2 }{ n }^{ 2 }+{ A }_{ 3 }{ n }^{ 3 }+\cdots. \]
Let's equate this function to our original summation:
\[ 1 + 2 + 3 + \dots + n = { A }_{ 0 }+{ A }_{ 1 }n+{ A }_{ 2 }{ n }^{ 2 }+{ A }_{ 3 }{ n }^{ 3 }+\cdots. \qquad (1)\]
Now we want to simplify this equation. There's no easy way to simplify this because the left hand side is not a proper expression. We first need to express the left hand side as a proper eexpression.

The easiest way to do this is to compare the above expression to the summation \(1 + 2 + 3 + \dots + n + \left( n+1 \right) \).